The correlation between two asset classes is a measurement of how much the two are moving in step.
Put simple, asset classes with a high positive correlation will move up and down very much alike. On the other hand will asset classes with low correlations not move in lock step.
Correlation is always calculated on past data, so there is no guarantee that the number will stay absolutely the same.
Let us illustrate the above said:
The S&P 500 Index has a very high correlation to the DJIA with 0.97. This is only logical, as all of the 30 DJIA Stocks are by definition the biggest entities of the SP500 Index.
On the other hand is the correlation between the price of gold, and the SP500 only 0.06; showing that the two have not moved into the same directions very often.
Asset classes with small or even negative correlations make for a smother ride, as the ups and down's of one asset class should be evened out by the down's and up's of the other. It also offers rebalancing opportunities.